TRIGONOMETRY

MATHEMATICS PROJECT

SUBMITTED BY HK MADHULIKA X B 38

INTRODUCTIONThe word trigonometry is derived from the Greek words tri meaning three, gon meaning sides and metron meaning measure. In fact, Trigonometry is the study of relationships between the sides and angles of a triangle. Most of the technologically advanced methods used in Engineering and Physical Sciences are based on trigonometrical based concepts. It is fundamental to astronomy and navigation. It is also the foundation of the practical art of surveying. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn , essential to many branches of science and technology.

HISTORYTrigonometry is a field of mathematics first compiled in 2nd century BC by the Greek mathematician Hipparchus. In 1595, an influential work on trigonometry was published which may have coined the word "trigonometry". Early study of triangles can be traced to the 2nd millennium BC, in Egypt In Indian astronomy, the study of Egypt. , trigonometric functions flowered in the Gupta period, especially due to Aryabhata (6th century). period During the middle Ages, the study of trigonometry continued in Islamic mathematics, whence it was adopted as a s mathematics, separate subject in the Latin West. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th Enlightenment, 17thcentury mathematics (Isaac Newton and James Stirling) and reaching Isaac ) its modern form with Leonhard Euler (1748).

TRIGONOMETRIC RATIOS OF ACUTE ANGLES

0 sin cos Tan cosec Sec Cot0 1 0 Not Defined 1 Not Defined 1 2 2 1

30

45

60

901 0 Not Defined 1 Not Defined 0

We observe: As increases from 0 to90, sin increases from 0 to 1 and cos A decreases from 1 to 0. tan = cot = exceed 1, whereas the , The values of sin and cos never value of sec and cosec are always greater than or equal to 1. The value of cosec the reciprocal of sin The value of sec the reciprocal of cos The value of tan the reciprocal of cot Hence, these trigonometric relationships are verified : Sin = Cosec = Cos = Tan = Sec = Cot =

TRIGONOMETRIC RATIOS OF COMPLEMENTARY ANGLES

Two angles are said to be complementary if their sum equals 90. In ABC, right-angled at B, since A + C = 90, they form such a pair. We have: cosec A= sin A= cos A = tan A = sec A = cot A =1

TRIGONOMETRIC IDENTITIESIdentities are those equations that hold true for any value. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved. In ABC, right angled at B we have: (1) AB2 + BC2 = AC2 2 Dividing each term of (1) by AC ) i.e., (cos A) 2 + (sin A) 2 = 1 A 2 2 cos A + sin A = 1 (2) This is true for all A such that 0 A 90. Dividing eqn (1) by AB2

1 + tan2 A = sec2 A (3) This is true for all A such that 0 A 90. Dividing (1) by BC2